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In this note we shall prove some supplementary results of Jacobson [5] concerning Kronecker products of primitive algebras and those of P.M.I. algebras (that is, algebras with faithful minimal one sided ideals) and study their applications....

Let F be a commutative ring with unit element 1. A ring R is called an algebraic ring over F by Drazin, in his recent paper [2], when R is an algebra over F and every element x of R satisfies an equation of lower monic form

Several years ago, M. Ward and the author [4] began a study in abstract form of the ideal theory of commutative rings. Since it was intended that the treatment should be purely ideal-theoretic, the system which was chosen for the study was a lattice with a commutative multiplication. For such multiplicative lattices, analogues of the Noether decomposition theorems for commutative rings were formulated and proved. However the theorems corresponding to the deeper results on the ideal structure of commutative rings were not obtained; the essential difficulty being the problem of formulating abstractly the notion of a principal ideal. This difficulty occurred in a mild form in treating the Noether theorem on decompositions into primary ideals. In fact, in the above mentioned paper, a weak concept of “principal element” was introduced which sufficed for the proof of the decomposition theorem into primaries. Nevertheless, the definition had serious defects and it was immediately obvious that it was not adequate for the further development of the abstract theory.

In this short note we shall always assume that a ring A is a commutative Noetherian ring with unit element and A-modules are unitary. We use [2] as a reference source for homological algebra....

Let J be a directed set and let {Λj, φij} be a direct system of rings indexed by J and with limit Λ. Let {Aj, φij} be a direct system of groups indexed by J. Assume that each Aj is a left Λj-module and that φij(λa) = φij(λ) φij(a) for each λ ∈ Λj, a ∈ Aj. Then the limit A of (Aj, φij) is a left Λ-module.

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.